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Roberts’ example and the AR-property of convex sets in non-locally convex linear metric spaces. (English) Zbl 1003.54501
The paper begins with some still open, well-known problems in topology (the AR-problem, Schauder’s conjecture and the Banach problem of whether every infinite-dimensional compact convex set in a linear metric space is homeomorphic to the Hilbert cube). Then, in Section 2 the following characterization of ANR-spaces is given: A metric space is an ANR if and only if there is a zero sequence of open covers \(\{{\mathcal U}_n\}\) of \(X\) together with a map \(f:{\mathcal K}({\mathcal U})\to X\) such that, for any map \(g:{\mathcal U}\to X\) satisfying \(g(U)\in U\), \(U\in{\mathcal U}\), and for any sequence \(\{\sigma_k\}\subset{\mathcal K}({\mathcal U})\) with \(n(\sigma_k)\) converging to \(0\), we have \(\text{diam}\{g(\sigma^0_k)\cup f(\sigma_k)\}\to 0\). Here, \(\{{\mathcal U}_n\}\) is a zero sequence if \(\text{mesh}({\mathcal U}_n)\to 0\), \({\mathcal U}= \bigcup\{{\mathcal U}_n; n\in\mathbb{N}\}\) and \({\mathcal K}({\mathcal U})= \bigcup\{N({\mathcal U}_n\cup{\mathcal U}_{n+1})\); \(n\in\mathbb{N}\}\), where \(N({\mathcal U}_n\cup{\mathcal U}_{n+1})\) is the nerve of the cover \({\mathcal U}_n\cup{\mathcal U}_{n+1}\) equipped with the Whitehead topology. Also, \(n(\sigma)= \sup\{n\in\mathbb{N}; \sigma\in N({\mathcal U}_n\cup{\mathcal U}_{n+1})\}\) for each \(\sigma\in{\mathcal K}({\mathcal U})\). In Section 3 some results and questions concerning Roberts’s examples of compact convex sets with no extreme points are reviewed. The last section is devoted to the following theorem: Let \((X,\mu)\) be a normalized measure space and let \(\phi\) be an Orlicz function. Then \(L_\phi(\mu)= \{f: X\to\mathbb{R}\), \(\int_X\phi(|f|)d\mu< \infty\}\) with the \(F\)-norm \(\|f\|= \int_X\phi(|f|) d\mu\) contains a needle point space if and only if \(\mu\) is not purely atomic.
Reviewer’s comment: It is shown that the above-mentioned characterization of ANR’s (Theorem 2.1 in the paper under review) implies the Dugundji theorem that every convex set in a locally convex space is an AR. But it should be noted that in the authors’ proof of Theorem 2.1 some results from the monograph by S. Hu [Theory of retracts (1965; Zbl 0145.43003)] are used and the proof of these results (as given in Hu’s book) are based on the same technique developed by Dugundji for proving his famous theorem.
MSC:
54E35 Metric spaces, metrizability
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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